Problems of Thermal Conduction in One Dimension in Bodies with Varying Conductivities by Duhamel's Method

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Problems of Thermal Conduction in One Dimension in Bodies with Varying Conductivities by Duhamel's Method

Sawada, M.

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T1t1e:

Author: Reference: Trans1a.tor:

NATIONAL RESEARCH COUNCIL OF CANADA Technical Trans1at10n 896

Problems of thermal conduct10n 1n one d1mens10n 1n bod1es w1th vary1ng conductivities by Duhamel's methOd

Masao Sawada

J. Soc. Mech. Engrs., Japan, 36 (190): 127-130, 1933 K. Shimizu

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PREFACE

One of the projects of the F1re Sect10n of the D1v1s1on of Bu1ld1ng Research concerns the development of a method for calculat1ng the f1re endurance of bU1ld1ng elements. The ma1n theoretical problem 1n th1s ,work 1s that the

class1cal solut1ons of the Four1er equat1on, based on con-stancy of propert1es, cannot be used because of the w1de var1at1ons 1n temperature 1nvolved 1n th1s exper1mental work.

Three papers by Masao Sawada that deal w1th problems of the heat conduct1on encountered when the heat capac1ty and thermal conduct1v1ty of the so11d are var1able, have been translated and 1ssued as NRC Techn1cal Translat10ns

Nos. 895, 896 and 897. From the work of Sawada 1t 1s seen

that the more perfectly the mechan1sm of heat conduct1on w1th1n the so11d 1s approached, the more l1m1ted the ap-p11cab1l1ty ·of the method becomes, as far as the shape or the so11d or the boundary cond1t1ons are concerned.

Although the 1ntroduct1on of var10us numer1cal methods and computer techn1ques largely reduce the pract1cal value or such analyt1cal methods, there are certa1n f1elds where they are 1nd1spensable.

Ottawa,

September 1960

R.P. Legget, Director

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PROBLEMS OF THERMAL CONDUC':'ION IN ONE DIMENSION IN BODISS WITH VARYING CONDUCTIVITIES BY DUHAr1EL'S ME'J'HOD

Abstract

In the present article, solutions are derived of probiems of thermal conduction in one dimension in bodies with varying thprmal conductivities by Duhamel's method in cases where the surface temperatures are controlled by

periodic and non-periodic functions of time. The author

w1shes'that the readers of the present article will refer to his article "Problems of thermal conduction in one dimension in bodies with varying conduct1vities" (J. Soc.

Mech. Engrs., ｾ｡ｰ｡ｮＬ 35 (177), 10).

1. Integration by Duhamel's Method

If A = ａｏｲｾＬａＯｰｾ = _{a 0}2 r 2

- P, m1

=

0 (flat slab), 1 (circular

cylinder), 2 (sphere); k

=

2 or 3/2 (here 2), m1

+ ｾ - 1 = m,

n

=

m/p ｾ 0, and further of a fractional type, the temperature is

v=exp.(-t"l!ao:lp'a,'/)-Ji:,,(U!a,VrP)/(v'YP)",

71.=,,-.,/,....

(constant flow), (1)

When A= AOemx, A/p.y = _{a 0}2ePX, m'

=

0, n = m/p,

eX is to be substituted in (1) in place of r.

When A= AO(C

±

ｲＩｾＬａＯｰｾ

=

a

0

2(c _ｾ _r)2-p,

If m' = 0, c

±

X is to be substituted in (1) in place of r.

If m' 4: 0, and ｾＬ p and m assume special values, a solution may be

obtained if the difference between l/r and l/(c + r) is not too

great and therefore l/r ｾ

1/(0

+ r).

For brevity, let the double cylindrical function and the double trignometric function, l\'h 1ch satisfy the boundary conditions, be

represented by Un(r)

=

ｕｮＨＲ｡ｳｾＩＪＮ For instance, if u

=

ｾＱ - ｾＯｲｭ

(5)

is 1ntegrated:

where

-4-ｩｾ＾ＫｭＭＱｃ - 1IJ'U,.{t:)dr=-('IP+'2Q)Ipa.

p=1

rp+mU,,+I(d

I::.

Q=!Un-ldl::.

= 0, l/h_a ｾ [Vi] + [v] - a = 0, where

r a r a

v'

=

ov/or

and the same for others. P and Q will be used in what follows.

Consider a general case as a boundary condit1on.

Table I ｾｮ､ Fig. 1.

l/hi • [vI ] + b - [v]

ri ri

[v]t=O = f(r) and further

See Table I r; t V( ••• )=UI.)+W(, ••) l/Il,'(v'J.;= (v).;-b l/h.·(v') ••= a - (v) •• (v).=.=f(r) l/Il;'(u'J.,=u,-b l/h., (u') •• =a-u. u= (v).:. l/Il,'(w'J.,=w, l/h. '(w')••=-w. (w).=.=f(r)-u respectively. If ha

=

0 Fig. 1

If a

=

1jI(t), and b = ｾＨｴＩＬ [w]t=O = 0 .: u = f(r).

If _{l/ha = 0 and l/hi = 0, [v]r}

a c a or 1jI(t); [v]r1

=

b or ｾＨｴＩｊ

and h 1 = 0, [v']r = 0, [vl]r = 0, respectively.

a i

Further, when a and b are functions of t, a solution is

obtain-ed for the case when they are always constants. The purpose of the

present article 1s to apply Duhamel's method· to the solution so obtained.

Generally, in order to solve cases where surrounding tempera-ture is a function of time, in addition to the above, other methods, such as that of ordinary integration, Riemann, Stokes, Angstrom·,

Carslaw· and Bromwich·· are available. However, it seems to the

•

Carslaw. Conduction of heat. pp. l'l, 68, 42 and 210-224 •

••

Bromwich. Phil. Mag. v. 37, 1919.

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-5-author that Duhamel's method ls the most convenient when the lnltlal temperature inside the body ls a function of dlmenslon. The calculation ls shown below for convenlence:

(1) When the change ls perlodlc.· Let Ps = - (a

_o

P8

s)2.

a=¢(t)=¢o+!C¢/coskwt+¢/'sinkwlj, b=r/J(t)=r/Jo+t(l/>/coskwt'+r/J/'sinkwtJ

,,=\ " = 1 ·

f¢(I1)'ｯｾﾷＭｯＧｴｬｬＱ = f1.Gk(t) ·.. ·.. ·.. · (1ll

f¢(I1)' O,e"It-Oldl1=f1.H

k(t) (13)

wher-e _{G"(t)-¢o(er-·'-1)/f1.+E,,,.,,{({1.·}sin(kwHB",.,,)+kw·cos(kwt+B ,,)J

ＫｾＢＨｻＱ ••sin B....,,+kw·cosB",.J}/B.'+(kw)' , H,,(t) ..r/Jo(C'·'-I)/I3.+ E".,,{CI1.·sin(kwH B".,,)+kw •cos(kwHB".,,)J

. +C',c({1.'sin B".,,+kw·cos B".k)}/P.'+(kw)'

E",."=(¢{J+¢"my',, B",.,,=tan-_'¢/'/¢/, E"."=(r/J/j+r/J"II2)",, B"",=tan- 'ｲＯｊＢｉｉＯｾＯ

(11) When the change ls non-perlodlc. ｾｳ the same as before.

L'

¢ ,,(I1 ) 'ｯｬｾＨｃＭｏＩ､ｬＱＧＢ 11.' 'F,,'C',,-'F,,(t) (1.) L'r/J,,(11) •ｯｬ･ｾﾷＱｬＭＰＩ､ｬＱ =11• • _" •C'.,-CD,,(t) (15) where 10 'F,,=¢o/I1.+ l'k!'¢,,/I1."+t, ,,=\

"

CD,,=r/Jo/I1.+ l'k! • r/JJI1.k+l, ,,=\ 'F,,(t) ...ﾢＬＬＨｴＩＫﾢＢ｟ｉＨｴＩＯｻＱＮＫﾢｈＨｴＩＯｉＱＮｾＫ +¢I(t)/f1."-1 +k! •¢JI1.", CD,,(t)=r/J,,(t)+r/J,,_I(t)/I1.+ r/J"_2(t)/11.2

+

+r/Jl(t)/I1.k-1+k! • r/J.jl1.". ..., ...• . ,

ｾｫ｟ＱＨｴＩＧ ¢k_2(t), •••••¢,(t) take the same form as

V

_{k_}₁( t ) ,

'k_2(t), ••••• ',(t). In what follows, references wlll be made to

Gk(t), Hk(t), Wk(t), Wk, tk(t) and ｾｫＮ

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-6-2. Problems of Thermal Conduction

[10] _l/hi • [av/ar] _{= [v]r} - b,

i ri _i

and further [ v ]t= 0 = f (r ) •

( I) When a and b are fixed.

Solution: where

l/h_a • [av/ar]

r a

=

a - [v]r 'a

Thus the quantity of heat flow can be calculated.

(II) When a and b are periodic functions. Refer to 1, (i).

Solution: where

(I)

(III) When a and b are non-periodic functions. Refer to 1, (ii).

Solution: v=

1, 1m.,

a.' U"(r){CXXfDk(t)J+CYXWk(t)J+(a,,pa.)2e-ao·P·d••tCCXX(1)kJ+CY)(WkJJ} + CzJ ...(211

ＮｾｬｫｾＱ

In (II) and ＨｾｉｉＩＬ the effect of exp.(-a02p2as2t) will diminish

with time and finally become non-existent. The case will be

further simplified if f(r)

=

0 or a constant.

[2°] [v]

=

b , l/h • [av/ar]

=

a -

lvl, '

[v]t=O

=

f(r).

ri a _{r a} a

When a and b are constants.

•

Sawada, Masao: J. Soc. Mech. Engrs., Japan, 35 (177), 14 with

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-7-Solu tion : v=u+ｉ＾ｾｾｾｾ . aｾＮ exp (-aｾｉ＾ｾ｡ｾｴＩ . U.(r){ froyP+",-1n ). 11(r)d ('P+ Q)!'--;:;-} (2 )

ｲＮｾｉＢ . Or' ,,- Jr; /\1' u,,_ 1'-'1 '2 rae ,

where u=

'I

-'2/1'''',

'I

=Cb(m/iz:r.m+1-r.-"')+ari-"'J/q, '2=(a -b)/q, q= m/h."."'·1+(ri-'" -1'.-"'),

ｕＬＬＨｴｊ］ｃｊＬＬＨ､ﾷｊＭＢＨｾｩＩＺｉＭＬＬＨｲＩﾷｊＢＨｾｩＩｊＯｖｲＢＧＬ as is a positive root of ｃｕＬＬＧＨｾＩｊｲＮＫｨＮｕＬＬＨｾＮＩ］ｯ •. lll.=ｻｃｰ｡ＮｙｲＮｬＧＫｨＮｾｲＮＲ - mh.r.Jr."'CU,,(r.)Y -(psin nn:/lIytl

•

Further " fr.

8v/f}r=ｭＧＲＯｲＢＧＫＱＫｰＴｾＯｬｊ ••,｡Ｎｾﾷ 1'1'-1 •U"u(r)·expo ( - alp2a.2t) . Jr, yP''''-ICj{r)- uJU,,(r)dr ...(2,)

(II) a and b are ｰ･ｲＱｯｾｩ｣ functions. Refer to 1, (i).

Solution: where

ｶ］｡ｯｾｰＺｾ ·l;Illl•.a.3

•U,,(tJ{CXJC(G(t)J+CY)(H(t)J }+CZJ (27)

X=C(m/h.r."'H -r.-"')p- QJ/q,_. Y=Cri-"'P+QJ/q,_. Z=ｐＺｾｬｬｬ •.a.2•exp.( - ｡ｯＲｰｾＬＧＱＬＮｾｴＩﾷ U,,(r)ｩｾﾷ yP+"'-'.f(r)· U,,(r)dr

Solution:

(III) a and bare non-per1odic funct1ons. Refer to 1, (ii).

v=ｰｾＮｾＮｫｾＧｩｊｊＮﾷ a.· U,,(d{CX)((1Jk(t)+(aopa.)2 . (1Jk·exp.( - ｡ｯＲｰｾ｡ＯｴＩｊ

ＫｃｙｊｃｾｫＨｴＩＫＨ｡ｯｰ｡ＮＩＲＮｾｫﾷ exp.( - ｡ｯＲｰｾ｡ＯｴＩｊｽＫｃＲＧｊ (25)

OOJ CvJr,=b, CvJr.=a, CvJt=J=}1.r)

(I)

When a and b are constants.

Solution: where

Further

ｾＡ］ U+pt'JJ•• a.2•ｾｸｰＮＨ - alp2a/t)· (In(dirOrl'+",-IC/(r)- uj U,,(r)dr (2p )

.=1

11='1- '2/1'''',

'I

=(ari-"'- br.-"')/q, '2=(a- b)/q, q=ri-"'- r.-rn,

ｕＬＬＨｲＩ］ｃｊＬＬＨｾＩ ·J-,,(ri)-ｊＭＬＬＨｾＩ ·J,,(ri)J/\/,;m, as is a posi ti ve root of u,,(r.)=o m.=(n:/sin nn:)2/(O.2-1), O.=J,,(r.)/J,,(t.)=J-ll(r.)/J-,,(r,i).

8v/8r=mi-dr":"

+

ｰＺｾｭ ••｡Ｎｾﾷ exp.( - ｡ｯＲｰｾ｡ＯｴＩﾷ U"+I(r)',.p-I

x

{I:·

yP+"'-'.f(r)· U,,(r)dr - (rJP+ ,IQ)!pa.} (21_s)

(II) a and b are periodic functions. Refer to 1, (i).

Solution:

v=(aop)2!ＮＱ［ｾｾ

..

｡Ｎｾﾷ U,,(d{exJcG(t)J+CYJCH(t)j}+(ZJ (211 )

.=1 k=1

X= -(r.-rnP+Q)/q, Y=(ri-mP+QlJi;,

z= p

1'13.·

a/·exp.( - ｡ｯＲｰｾ｡ＯｴＩﾷ Un(t) fr·,Hm-'!\r). U,,(I:)dr

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-8-(III) a and b are non-periodic functions. See 1, (ii).

Solution:

v=

t·

ｾｾ

..

a.' Unk){CXJC(Ok(/)+(aopa.r· (Ok'exp.] - ｡ｯｾｰｾ｡Ｏｬｮ

+CYJCｉｦＧｫＨＯＩＫＨ｡ｯｰ｡ＮＩｾﾷ If'k'exp.( - ｡ｯＲｰｾ｡ＯｴＩｊｽＫ CZJ (2I t )

In the preceding sections, the

cooling ｾｩｴｨ the boundary condition

author dealt with the case of

l/h • [v']

=

[V]r - b ,

i r_i i

with particular reference to 3 cases,

l/h

_a

• [v

'J

= a -

l vl

r a

r x

hi = 0,

Ilb

_{i =}

Ilb

_a

=

0. There are cases in which hi =

°

and hi =

h_a = 0. Since solutions are readily available in these cases, they_.

will not be dealt with in the present paper.

Generally, the consideration of the case of heating or cooling gives rise to the fOllowing:

±l/h,·(v'Jrj - (V)rj- 6, ±l/ha •CV')r.=a-(vJr.

where hi = 0, _{finite or infinite; ha} = 0, finite or infinite;

b = 0, constant or ¢(t) j a = 0, constant or vet) j

b

=

a or b

+

a; [v]t=O

=

0, constant or fer).

Further, the order n of the function ｕｮＨｾＩ of the solution can

be an integer or a fraction, positive or negative. Thus, various

combinations of these possibilities give rise to varied cases.

In the present article solutions were obtained by Duhamel's method

only. However, the author wishes to mention that, when

+l/ha[V']r

a

=

a - [v]ra; a

=

0, constant or wet); [v]t=o = 0, a

solution can be obtained, as in the case of homogeneous bodies, by Bromwich's method.